14-240/Classnotes for Monday September 22

DBN: Classes: 2014-15: 14-240 / Navigation Welcome to Math 240! (additions to this web site no longer count towards good deed points) # Week of... Notes and Links 1 Sep 8 About This Class, What is this class about? (PDF, HTML), Monday, Wednesday 2 Sep 15 HW1, Monday, Wednesday, TheComplexField.pdf,HW1_solutions.pdf 3 Sep 22 HW2, Class Photo, Monday, Wednesday, HW2_solutions.pdf 4 Sep 29 HW3, Wednesday, Tutorial, HW3_solutions.pdf 5 Oct 6 HW4, Monday, Wednesday, Tutorial, HW4_solutions.pdf 6 Oct 13 No Monday class (Thanksgiving), Wednesday, Tutorial 7 Oct 20 HW5, Term Test at tutorials on Tuesday, Wednesday 8 Oct 27 HW6, Monday, Why LinAlg?, Wednesday, Tutorial 9 Nov 3 Monday is the last day to drop this class, HW7, Monday, Wednesday, Tutorial 10 Nov 10 HW8, Monday, Tutorial 11 Nov 17 Monday-Tuesday is UofT November break 12 Nov 24 HW9 13 Dec 1 Wednesday is a "makeup Monday"! End-of-Course Schedule, Tutorial F Dec 8 The Final Exam Register of Good Deeds Add your name / see who's in!

Polar coordinates:

• [math]\displaystyle{ r \times e^{i\theta} = r \times cos\theta + i \times rsin\theta }[/math]
• [math]\displaystyle{ r_1 \times e^{i\theta_2} = r_1 \times (cos\theta + sin\theta }[/math]

The Fundamantal Theorem of Algebra: [math]\displaystyle{ a_n \times z^{n} + a_n-1 \times z^{n-1} + \dots + a_0 }[/math] where [math]\displaystyle{ a_i \in C }[/math]and[math]\displaystyle{ a_i != 0 }[/math] has a soluion [math]\displaystyle{ z \in C }[/math] In particular, [math]\displaystyle{ z^{2} - 1 = 0 }[/math] has a solution.

• Forces can multiple by a "scalar"(number).

No "multiplication" of forces.

Definition of Vector Space: A "Vector Space" over a field F is a set V with a special element [math]\displaystyle{ O_v \in V }[/math] and two binary operations:

• [math]\displaystyle{ + : V \times V -\gt V }[/math]
• [math]\displaystyle{ \times : V \times V -\gt V }[/math]

s.t.

• [math]\displaystyle{ VS_1 : \forall x, y \in V, x + y = y + x }[/math].
• [math]\displaystyle{ VS_2 : \forall x, y, z \in V, x + (y + z) = (x + y) + z }[/math].
• [math]\displaystyle{ VS_3 : \forall x \in V, x + 0 = x }[/math].
• [math]\displaystyle{ VS_4 : \forall x \in V, \exists y \in V, x + y = 0 }[/math].
• [math]\displaystyle{ VS_5 : \forall x \in V, 1 \times x = x }[/math].
• [math]\displaystyle{ VS_6 : \forall a, b \in F, \forall x \in V, a(bx) = (ab)x }[/math].
• [math]\displaystyle{ VS_7 : \forall a \in F, \forall x, y \in V, a(x + y) = ax + ay }[/math].
• [math]\displaystyle{ VS_8 : \forall a, b \in F, \forall x \in V, (a + b)x = ax + bx }[/math].

Scanned Lecture Notes by AM

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Scanned Lecture Notes by Boyang.wu

File:W31.pdf